Abstract: Hexagonal Hierarchies and Close Packing of the Plane: Overview--
A scatter of points, spread evenly across the plane, may take on a variety of configurations: two simple regular lattices involve points that suggest squares or equilateral triangles. If one wishes to consider circular buffers around each point, then these buffers may overlap or be widely spaced. A natural issue to consider is to provide some sort of maximal coverage of the plane by the buffers: to provide a "close packing" of the plane by circles. Gauss (1831/40) proved that the densest lattice packing of the plane is the one based on the triangular lattice. In 1968 (and earlier), Fejes-Toth proved that that same packing is not only the densest lattice packing of the plane but is also the densest of all possible plane packings. If one thinks, then, of the circles as if they were bubble foam, the circles centered on a square grid pattern expand and collide to form a grid of squares (Boys). The circles centered on a triangular grid pattern expand and collide to form a mesh of regular hexagons, like the cells in a slice of the honeycomb of bees (de Vries). The theoretical issues surrounding tiling in the plane are complex; even deeper are those issues involving packings in three dimensional space. The reader interested in probing this topic further is referred to the Bibliography at the end of this document. Interpretation of the simple triangular grid has range sufficient to fill this document and far more.
Classical Urban Hexagonal Hierarchies--
One classical interpretation of what dots on a lattice might represent is found in the geometry of "central place theory" (Christaller, Lösch). This idea takes the complex human process of urbanization and attempts to look at it in an abstract theoretical form in order to uncover any principles which might endure despite changes over time, situation, cultural tradition, and all the various human elements that are truly the hallmarks of urbanization. Simplicity helps to reveal form: models are not precise representations of reality. They do, however, offer a way to look at some structural elements of complexity.

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